Recently, in [1]} was proved that the same conditions implies also that \(a(D)\in {\mathcal {L}}(L^p_w(X), L^p_w(Y))\)  for every \(w\in A_p\)  and every \(p\in (1,\infty )\) ; see
[1]}. Moreover, it is readily seen from the proof that for every \({\mathcal {W}}\subset A_p\)  with \(\sup _{w\in {\mathcal {W}}} [w]_{A_p}<\infty \)
\(\sup _{w\in {\mathcal {W}}}\Vert a(D)\Vert _{{\mathcal {L}}(L^p_w(X), L^p_w(Y))} < \infty .\) 
